Regression discontinuity I am analyzing some Stata code for a regression discontinuity analysis. The results of this analysis are presented in table 5  of the Online appendix to “The Effects of a Universal Child Benefit...”. I first need to understand the code below, and then to make a twist (e.g. estimate the regression discontinuity with and without parametric estimation, or any other twist that I can come up with for this data).
Does anyone understand if everything is done already or if there is some possibility to innovate on it?
post is a dummy variable where 1 implies that the person has the treatment. m is month when the baby was born and determines the cutoff point whether or not the person received the treatment.
***********      Employed   *****************************/
xi: reg work2 post i.post|m i.post|m2 age age2 age3 immig primary hsgrad univ sib pleave i.q if m>-10 & m<9, robust 
xi: reg work2 post i.post|m age age2 age3 immig primary hsgrad univ sib pleave i.q if m>-7 & m<6, robust 
xi: reg work2 post i.post|m age age2 age3 immig primary hsgrad univ sib pleave i.q if m>-5 & m<4, robust 
xi: reg work2 post age age2 age3 immig primary hsgrad univ sib pleave i.q if m>-4 & m<3, robust
xi: reg work2 post if m>-3 & m<2, robust 
xi: reg work2 post age age2 age3 immig primary hsgrad univ sib pleave i.q if m>-3 & m<2, robust
xi: reg work2 post m m2 age age2 age3 immig primary hsgrad univ sib pleave i.n_month i.q, robust cluster(m)

 A: Here's how I like to play with discontinuities:
Discontinuities can be estimated using hinge functions. Take the simple case of a linear relationship that jumps discontinuously at some point. The model for a straight line is (or ought to be :) familiar (leaving out error):
$$y = \beta_{0} + \beta_{x}x$$
If we want to add a discontinuous jump to this line (i.e. same slope, but with some constant shift at some point), we can instead estimate this model plus the simplest hinge function:
$$y = \beta_{0} + \beta_{x}x + \beta_{\text{c}}\max(\text{sign}(x - \theta),0)$$
Here $\theta$ is the value of $x$ at which the discontinuous jump happens, and $\beta_{\text{c}}$ is the magnitude and direction of that jump. But what do the values of $\max(\text{sign}(x - \theta),0)$ look like? Well they look like $0, 0, 0, 1, 1, 1$ with the $1$s corresponding to $x > \theta$. (There are other ways to specify such a jump rather than using the hinge function as I have done here—e.g. $\beta_{\text{c}}[\text{sign}(x-\theta)+1]/2$—but sit tight and watch what happens to the hinge function with more complicated models.)
How would one estimate this in Stata? By using nl (nonlinear least squares regression):
nl (y = {b_0} + {b_x}*x + {b_c}*max(sign(x-{theta})),0) )

The results of such a model would include four parameter estimates: b_0, b_x, b_c, and theta. If you have a good idea about where the discontinuous jump is, you can tell Stata's algorithm to start looking for values of theta. Here I have done so (only for the first instance of the {theta} parameter) for the value 8.5:
nl (y = {b_0} + {b_x}*x + {b_c}*max(sign(x-{theta=8.5}),0) )

As hinted at earlier, the hinge function is quite versatile. For example, to estimate a line which changes slope without a jump:
$$y = \beta_{0} + \beta_{x}x + \beta_{\text{c}}\max(x - \theta,0)$$
Here values of $\max(x - \theta,0)$ look like $0, 0, 0, 0, 1, 2, 3$ with $>0$ values corresponding to values of $x > \theta$ (assuming $x$ is measured in integer values).
We could likewise model a quadratic effect of $x$ that only begins at $\theta$:
$$y = \beta_{0} + \beta_{x}x + \beta_{\text{c}}\max(x - \theta,0)^2$$
And we can combine these, for example to produce discontinuous jumps, in addition to changes in slope:
$$y = \beta_{0} + \beta_{x}x + \beta_{\text{c}0}\max(\text{sign}(x-\theta),0) + \beta_{\text{c}x}\max(x - \theta,0)$$
And for a discontinuous jump permitting a fully quadratic relationship of $x$:
$$y = \beta_{0} + \beta_{x}x + \beta_{\text{c}0}\max(\text{sign}(x-\theta),0) + \beta_{\text{c}x}\max(x - \theta,0) + \beta_{\text{c}x2}\max(x - \theta,0)^2$$
